Finite Group Databasebeta

← #29833885

Subgroups — #29833885

Order 15616.

1098 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840579C211yesmaximal, characteristic, complemented
2C2×#16840202C211yesmaximal, characteristic, complemented
3C2×#16840192C211yesmaximal, characteristic, complemented
4#21447757C212yesmaximal, complemented
5#21447752C212yesmaximal, complemented
6#21447740C212yesmaximal, complemented
7#21447735C212yesmaximal, complemented
8#21447529C211yesmaximal, characteristic, complemented
9C2×#16839565C211yesmaximal, characteristic, complemented
10#21447433C211yesmaximal, characteristic, complemented
11C61×#3061C211yesmaximal, characteristic, complemented
12C2×#14519824—21no—
13#16840579—24no—
14#16840579—24no—
15C2×#14519810—21no—
16C2×#14519810—22no—
17C2×#14519799C2212yes—
18C2×#14519799C2211yescharacteristic
19C22×Dic244C2212yes—
20C22×Dic244C2211yescharacteristic
21#16840202—24no—
22#16840192—24no—
23C2×#14519723C2212yes—
24C2×#14519723C2211yescharacteristic
25C2×#14519721C2212yes—
26C2×#14519721C2211yescharacteristic
27C2×#14519720C2211yescharacteristic
28C2×#14519718C2211yescharacteristic
29C2×#14519717C2211yescharacteristic
30C22×#11758185C2211yescharacteristic
31#16839565C2212yes—
32#16839565C2212yes—
33#16839565C2212yes—
34#16839562C2212yes—
35C22⋊#11758185C2212yes—
36C22⋊#11758185C2212yes—
37C2×#135×C61C2211yescharacteristic
38C2×#131×C61C2211yescharacteristic
39C61×#537—21no—
40C61×#468—22no—
41C61×#467C2212yes—
42C61×#466C2212yes—
43C2×#107×C61C2211yescharacteristic
44C61×#417—21no—
45#14519824—28no—
46#14519824—44no—
47C2×#11758222—22no—
48C2×#11758222—21no—
49C2×#11758224—42no—
50C2×#11758224—41no—
51C22×Dic122—22no—
52C22×Dic122—21no—
53C2×Q8×D61—41no—
54#14519810—28no—
55#14519810—216no—
56#14519799—22no—
57#14519799—24no—
58#14519799—24no—
59#14519799D414yescomplemented
60C2×Dic244—22no—
61C2×Dic244—21no—
62C2×Dic244—22no—
63C2×Dic244—24no—
64C2×Dic244—24no—
65C2×Dic244D414yescomplemented
66C2×Dic244C2311yescharacteristic
67C2×Dic244C2312yes—
68C2×#11758190—22no—
69C2×#11758190—21no—
70C2×D488C2311yescharacteristic
71#14519723—22no—
72#14519723—24no—
73#14519721—22no—
74#14519721—24no—
75#14519720—22no—
76#14519718—22no—
77#14519717—22no—
78C2×#11758185—22no—
79C2×#11758185C2312yes—
80C2×#11758185C2311yescharacteristic
81C2×#11758185C2311yescharacteristic
82C2×#41×C61—21no—
83C2×#41×C61D411yescharacteristic
84#135×C61—22no—
85#135×C61—22no—
86C2×Q16×C61C2311yescharacteristic
87C2×SD16×C61—21no—
88C2×D8×C61C2311yescharacteristic
89#131×C61—22no—
90#131×C61—22no—
91C2×M16×C61D411yescharacteristic
92C2×M16×C61D412yes—
93C22×C8×C61C2312yes—
94C22×C8×C61C2311yescharacteristic
95C22⋊Q8×C61—22no—
96C22⋊Q8×C61—41no—
97#121×C61—22no—
98#121×C61—41no—
99#121×C61—42no—
100C2×C4⋊C4×C61—21no—
101#107×C61—21no—
102#107×C61C2312yes—
103#103×C61—22no—
104#102×C61—22no—
105#41×C61—22no—
106#41×C61—22no—
107#41×C61—42no—
108#41×C61—42no—
109C2×Q8×C61—21no—
110C2×Q8×C61D811yescharacteristic
111C2×D4×C61—21no—
112C2×D4×C61—41no—
113C2×D4×C61—42no—
114C2×D4×C61D812yes—
115C2×D4×C61D811yescharacteristic
116C22×C4×C61—41no—
117C22×C4×C61—41no—
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2×D411yescharacteristic
120C2×C22⋊D61—42no—
121C2×C22⋊D61—41no—
122C22×Dic61—42no—
123C22×Dic61—41no—
124Q8×D61—48no—
125#11758224—416no—
126#11758224—48no—
127#11758222—28no—
128#11758222—24no—
129#11758222—44no—
130#11758222—42no—
131C2×D244—21no—
132C2×C4×D61—42no—
133C2×C4×D61—41no—
134C2×Dic122—28no—
135C2×Dic122—24no—
136C2×Dic122—22no—
137C2×Dic122—21no—
138C2×Dic122—44no—
139C2×Dic122—42no—
140C2×Dic122—41no—
141C2×Dic122—42no—
142Q16×C61—21no—
143Q16×C61—21no—
144SD16×C61—42no—
145D8×C61—21no—
146D8×C61—21no—
147M16×C61—22no—
148M16×C61—24no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61C2×D412yes—
152C2×C8×C61C2×D411yescharacteristic
153C2×C8×C61C2×D412yes—
154C2×C8×C61C2×D412yes—
155C2×C8×C61C2411yescharacteristic
156C4⋊C4×C61—22no—
157C4⋊C4×C61—41no—
158C4⋊C4×C61—42no—
159C22⋊C4×C61—42no—
160C22⋊C4×C61—42no—
161Dic244—28no—
162Dic244—24no—
163Dic244—21no—
164Dic244—22no—
165Dic244C2×D412yes—
166Dic244C2×D414yes—
167D488—21no—
168D488C2×D412yes—
169#11758190—28no—
170#11758190—24no—
171#11758185—21no—
172#11758185—21no—
173#11758185—22no—
174C23×C61—41no—
175C23×C61C2×D812yes—
176C23×C61C2×D811yescharacteristic
177C22×D61—41no—
178Q8×C61—22no—
179Q8×C61—41no—
180Q8×C61—41no—
181D4×C61—24no—
182D4×C61—22no—
183D4×C61—41no—
184D4×C61—41no—
185D4×C61—82no—
186D4×C61—162no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—42no—
190C2×C4×C61—41no—
191C2×C4×C61—82no—
192C2×C4×C61—161no—
193C2×C4×C61#12012yes—
194C2×C4×C61C2×D812yes—
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61AGL(1,ℤ/8)12yes—
197C2×C4×C61C22×D411yescharacteristic
198C22⋊D61—48no—
199C22⋊D61—44no—
200C2×Dic61—48no—
201C2×Dic61—41no—
202C2×Dic61—42no—
203C2×Dic61—44no—
204D244—22no—
205D244—41no—
206C4×D61—48no—
207C4×D61—44no—
208Dic122—24no—
209Dic122—22no—
210Dic122—48no—
211Dic122—41no—
212Dic122—42no—
213Dic122—44no—
214C8×C61—22no—
215C8×C61#12012yes—
216C8×C61C22×D411yescharacteristic
217C8×C61C22×D411yescharacteristic
218#33527—611nomaximal, Sylow
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—22no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61—81no—
227C22×C61#44712yes—
228C22×C61C22×D811yescharacteristic
229C2×D61—42no—
230C2×D61—81no—
231C2×D61—81no—
232C4×C61—22no—
233C4×C61—82no—
234C4×C61—81no—
235C4×C61#44712yes—
236C4×C61C2×#12011yescharacteristic
237C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
238Dic61—42no—
239Dic61—44no—
240C2×#578—611no—
241C2×#510—611no—
242C2×#508—611no—
243#3061D6111yescharacteristic, complemented
244#2189—612no—
245#2188—612no—
246#2186—612no—
247#2185—612no—
248#2139—611no—
249C2×#358—611no—
250#2113—611no—
251C2×C61—22no—
252C2×C61—42no—
253C2×C61—161no—
254C2×C61C2×#44711yescharacteristic
255C2×C61#341611yescharacteristic
256C2×C61#341711yescharacteristic
257D61—82no—
258C2×2−1+4—1221no—
259#578—1224no—
260#578—1224no—
261C2×#137—1221no—
262C2×#137—1222no—
263C2×#135—611no—
264C2×#135—612no—
265C2×#135C2×D6111yescharacteristic
266C22×Q16—611no—
267C22×Q16—612no—
268C2×#131C2×D6111yescharacteristic
269#537—21no—
270#510—1224no—
271#508—1224no—
272C2×Q32—611no—
273C2×Q32—612no—
274C2×SD32—611no—
275C2×SD32—611no—
276C2×SD32—612no—
277C2×D16—611no—
278C2×M32—611no—
279C22×C16—611no—
280#468—22no—
281#467C2×D6112yes—
282#466C2×D6112yes—
283C2×#107C2×D6111yescharacteristic
284#417—21no—
285#358—612no—
286#358—612no—
287#358—612no—
288#357—612no—
289C22⋊C16—612no—
290C22⋊C16—612no—
291C61#3352711yescharacteristic, complemented, Sylow
2922−1+4—1228no—
2932−1+4—2444no—
294C2×#41—21no—
295C2×#41—1222no—
296C2×#41—1221no—
297C2×#41—2441no—
298C2×#41—2442no—
299C2×#41C22⋊D6111yescharacteristic
300C22×Q8—1222no—
301C22×Q8—1221no—
302C22×Q8—2441no—
303#137—1228no—
304#137—12216no—
305#135—22no—
306#135—22no—
307#135—614no—
308#135—1224no—
309#135—1224no—
310#135—1222no—
311C2×Q16—614no—
312C2×Q16—612no—
313C2×Q16—611no—
314C2×Q16—1224no—
315C2×Q16—1224no—
316C2×Q16—1222no—
317C2×Q16—1222no—
318C2×Q16—1221no—
319C2×Q16C22×D6111yescharacteristic
320C2×SD16—21no—
321C2×SD16—1222no—
322C2×SD16—1221no—
323C2×D8—611no—
324C2×D8C22×D6111yescharacteristic
325#131—22no—
326#131—22no—
327C2×M16C22⋊D6111yescharacteristic
328C2×M16C22⋊D6112yes—
329C22×C8C22×D6112yes—
330C22×C8C22×D6111yescharacteristic
331C22⋊Q8—22no—
332C22⋊Q8—41no—
333#121—22no—
334#121—41no—
335#121—42no—
336C2×C4⋊C4—21no—
337Q32—1222no—
338Q32—1224no—
339SD32—1222no—
340SD32—1224no—
341SD32—1222no—
342D16—1222no—
343M32—1222no—
344C2×C16—612no—
345C2×C16—611no—
346C2×C16—611no—
347C2×C16—1222no—
348#107—21no—
349#107C22×D6112yes—
350#103—22no—
351#102—22no—
352#41—22no—
353#41—22no—
354#41—42no—
355#41—42no—
356#41—1228no—
357#41—1224no—
358#41—2444no—
359#41—24416no—
360#41—2448no—
361#41—2442no—
362C2×Q8—21no—
363C2×Q8—1228no—
364C2×Q8—1222no—
365C2×Q8—1221no—
366C2×Q8—1224no—
367C2×Q8—2444no—
368C2×Q8—2448no—
369C2×Q8—2442no—
370C2×Q8—2441no—
371C2×Q8—2442no—
372C2×Q8#1175819911yescharacteristic
373C2×D4—21no—
374C2×D4—41no—
375C2×D4—42no—
376C2×D4—1221no—
377C2×D4—2442no—
378C2×D4—2441no—
379C2×D4#1175819912yes—
380C2×D4#1175819911yescharacteristic
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—2442no—
384C22×C4—2441no—
385C22×C4—2442no—
386C22×C4—2441no—
387C22×C4C2×C22⋊D6112yes—
388C22×C4C2×C22⋊D6111yescharacteristic
389Q16—21no—
390Q16—21no—
391Q16—614no—
392Q16—612no—
393Q16—1228no—
394Q16—1222no—
395Q16—1221no—
396Q16—1224no—
397SD16—42no—
398SD16—1228no—
399SD16—1224no—
400D8—21no—
401D8—21no—
402D8—612no—
403D8—1221no—
404M16—22no—
405M16—24no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8D4×D6112yes—
409C2×C8D4×D6112yes—
410C2×C8C2×C22⋊D6112yes—
411C2×C8C2×C22⋊D6111yescharacteristic
412C2×C8C23×D6111yescharacteristic
413C4⋊C4—22no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C16—1221no—
419C16—1221no—
420C16—1222no—
421C23—41no—
422C23—2441no—
423C23C2×#1175819912yes—
424C23C2×#1175819911yescharacteristic
425Q8—22no—
426Q8—41no—
427Q8—41no—
428Q8—1224no—
429Q8—1222no—
430Q8—2448no—
431Q8—2441no—
432Q8—2442no—
433Q8—2444no—
434D4—24no—
435D4—22no—
436D4—41no—
437D4—41no—
438D4—82no—
439D4—162no—
440D4—1222no—
441D4—2448no—
442D4—2444no—
443D4—2441no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—82no—
449C2×C4—161no—
450C2×C4—2448no—
451C2×C4—2448no—
452C2×C4—2442no—
453C2×C4—2441no—
454C2×C4—2444no—
455C2×C4—2444no—
456C2×C4#1451974512yes—
457C2×C4C2×#1175819912yes—
458C2×C4C2×#1175819911yescharacteristic
459C2×C4#1451980212yes—
460C2×C4C22×C22⋊D6111yescharacteristic
461C8—22no—
462C8#1451974512yes—
463C8C2×D4×D6111yescharacteristic
464C8C2×D4×D6111yescharacteristic
465C22—21no—
466C22—21no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—4881no—
474C22—4881no—
475C22—2442no—
476C22#1683989512yes—
477C22C22×#1175819911yescharacteristic
478C4—22no—
479C4—82no—
480C4—81no—
481C4—2444no—
482C4—2442no—
483C4#1683989512yes—
484C4C2×#1451974511yescharacteristic
485C4C2×#1451980211yescharacteristic
486C2—22no—
487C2—42no—
488C2—161no—
489C2—4882no—
490C2C2×#1683989511yescharacteristic
491C2#2145896611yescharacteristic
492C2#2145897011yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3061C2
socleC22×C61C22×D8
hypercenter#3061D61
hypocenterC61#33527
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #3061

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840579C2semidirect
2. C2×#16840202C2semidirect
3. C2×#16840192C2semidirect
4. #21447757C2semidirect
5. #21447752C2semidirect
6. #21447740C2semidirect
7. #21447735C2semidirect
8. #21447529C2semidirect
9. C2×#16839565C2semidirect
10. #21447433C2semidirect
11. C61×#3061C2semidirect
59. #14519799D4semidirect
65. C2×Dic244D4semidirect
243. #3061D61semidirect
291. C61#33527semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168405792
C2×#168402022
C2×#168401922
#214477572
#214477522
#214477402
#214477352
#214475292
C2×#168395652
#214474332
C61×#30612
#3352761